One-electron self-interaction error and its relationship to geometry and higher orbital occupation Dale R. Lonsdale and Lars Goerigk

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One-electron self-interaction error and its relationship to geometry and higher orbital

occupation

Dale R. Lonsdale and Lars Goerigk∗

School of Chemistry, The University of Melbourne, Victoria 3010, Australia

Density Functional Theory (DFT) sees prominent use in computational chemistry and physics,

however, problems due to the self-interaction error (SIE) pose additional challenges to obtaining

qualitatively correct results. An unphysical energy an electron exerts on itself, the SIE impacts

most practical DFT calculations. We conduct an in-depth analysis of the one-electron SIE in

which we replicate delocalization eﬀects for simple geometries. We present a simple visualization

of such eﬀects, which may help in future qualitative analysis of the one-electron SIE. By increasing

the number of nuclei in a linear arrangement, the SIE increases dramatically. We also show how

molecular shape impacts the SIE. Two and three dimensional shapes show an even greater SIE

stemming mainly from the exchange functional with some error compensation from the one-electron

error, which we previously deﬁned [Phys. Chem. Chem. Phys. 22, 15805 (2020)]. Most tested

geometries are aﬀected by the functional error, while some suﬀer from the density error. For the

latter we establish a potential connection with electrons being unequally delocalized by the DFT

methods. We also show how the SIE increases if electrons occupy higher-lying atomic orbitals;

seemingly one-electron SIE free methods in a ground are no longer SIE free in excited states, which is

an important insight for some popular, non-empirical DFAs. We conclude that the erratic behavior

of the SIE in even the simplest geometries shows that robust density functional approximations

are needed. Our test systems can be used as a future benchmark or contribute towards DFT

development.

I. INTRODUCTION

It is commonplace to accompany or predict molecu-

lar or solid-state chemistry and physics with quantum-

chemical calculations. However, with the relative ease

anyone can conduct such calculations, alongside contin-

uing development in the ﬁeld, we must be mindful of

the capabilities and inadequacies of the methodologies

we use.

In the realm of molecular quantum chemistry we

can broadly designate two camps in which many pop-

ular computational methods lie: Wave Function Theory

(WFT) and Density Functional Theory (DFT).1,2 Short-

comings of the former are mostly a question of computa-

tional feasibility: including just enough electron correla-

tion to capture the essential physics governing the system

in question while getting the result in a timely fashion. In

contrast, DFT is generally computationally less intensive,

but retains drawbacks which diﬀer depending on which

of the applicable forms of DFT is under consideration.

Wildly popular is the Kohn-Sham DFT formalism2which

sacriﬁces computational eﬃciency for the reintroduction

of orbitals, thus yielding a level of guaranteed accuracy

for the electronic kinetic energy. Instead, the primary

shortcoming of DFT is ﬁnding accurate functional forms

for the exchange and correlation terms, which still re-

main elusive. In lieu of an exact solution to the prior two

terms, it becomes pragmatically justiﬁed to use density

functional approximations (DFAs) — all of which intro-

duce errors. It is the improper calculation of exchange

∗lars.goerigk@unimelb.edu.au

which leads to one of the most pervasive pitfalls of DFT:

the notorious self-interaction error (SIE).

For better or worse, routine computational chemistry

applications use DFT approaches — most of them suf-

fering from varying degrees of SIE,3,4 often unbeknownst

to mere DFT users. This error subtly, though some-

times dramatically, changes the quantitative result of ev-

ery practical DFT application and proves to be a barrier

to computational modelling. Past attempts to separate

and correct the SIE have had varying success,5which is

why one important aspect is to understand its empirical

nature so that the users and developers of DFT methods

can predict when their calculations have been inﬂuenced

by this spurious energy contribution.4,6–8

Dedicating research eﬀorts to the SIE is justiﬁed

due to its ability to cause qualitative errors. For

example in the case of the simplest hydrogen trans-

fer reaction, H2+H H+H2, many DFAs such

as BLYP9–11 drastically underestimate this transition

state energy,12 sometimes predicting other similar hydro-

gen abstraction reactions to be barrier-less processes.13

In such cases, ﬁnding the geometry of the transition

state is fruitless due to the SIE’s eﬀect on the po-

tential energy surface. Spurious proton transfers were

reported for organic acid-base co-crystals due to the

SIE/delocalization problem.14 Furthermore, there are is-

sues relating the SIE to applications including, but not

limited to, charge transfer,15–19 magnetic properties,20

spin-state splittings/energetics,21–25 transition state

energies,26 thermochemistry,3,27–29 halogen30,31 and

chalcogen bonding,31,32 halogen-bond dissocation,33 sol-

vated electrons,34 and dissociation of neutral species re-

sulting in unphysically delocalized electrons.35,36

One phenomenological description of the SIE’s man-

arXiv:2210.03386v2 [physics.chem-ph] 25 Nov 2022

ifestation is that it is an artiﬁcial delocalization of a

single electron over two or more nuclei at impossibly

large distances.8,34,35,37–39 The electron does not prop-

erly localize on one center and can instead have its neg-

ative charge stabilized over a longer distance. More for-

mally, the SIE comes from the expression for the classical

Coulomb integral:

J[ρ] = 1

2ZZ ρ(~r1)ρ(~r2)

~r12

d~r1d~r2,(1)

where ρis the electron density, ~r represents the spatial

coordinates of an electron, and ~r12 is the distance be-

tween two electrons. For a single electron the Coulomb

operator yields a non-zero value and leads to a repulsive-

self interaction energy between one electron and itself.

This term also exists in WFT, but is cancelled out at the

Hartree-Fock (HF) level by the corresponding exchange-

based self-interaction integrals. As exchange is approx-

imated in DFT, only an imperfect cancellation occurs,

leading to the SIE.

For as long as approximate exchange functionals have

been used, the SIE has been an issue which many have

sought to solve; for a recent review, see Ref. [8].

One of the most used explicit self-interaction correc-

tions (SICs) comes from the orbital-by-orbital removal

approach developed by Perdew and Zunger.40 This ap-

proach has limited success due to shortcomings asso-

ciated with the technique,5,41,42 though other avenues

to alleviate these issues are an ongoing source of re-

search: some examples include scaling43,44 and use of

Fermi-45,46 or complex-orbitals.47–49 In the solid-state

realm, the DFT+U correction50 is quite popular. How-

ever, simply correcting or removing the SIE is not nec-

essarily a straightforward improvement; SIE often mim-

ics important, non-dynamic electron correlation eﬀects

absent in conventional DFT calculations.51,52 Therefore,

one of the more straightforward solutions is the admix-

ture of exact Fock-exchange in a class of DFAs known

as hybrids.53 DFA exchange components can come from

the Local Density Approximation (LDA), Generalized-

Gradient Approximation (GGA), or meta-GGA (mGGA)

forms. Hybrids are commonly employed as a kind of

self-interaction correction, but using too much exact ex-

change inherits the problems associated with HF theory.

DFA development through benchmarking can cre-

ate statistically improvable functionals that perform

well across a range of test systems covering electronic

ground3,29,54–56 and excited states.18,57–64 Naturally, to

truly ﬁx the SIE and related issues we must also have ad-

vancement in satisfying constraints from ﬁrst principles:

see, Refs [65] and [66] for examples. Fixing DFAs from

a more fundamental level aims to solve problems like the

SIE in an elegant manner — perhaps leading to the next

generation of electronic structure methods. In lieu of this,

we can attempt to dexterously avoid and identify the SIE

through studies that characterize its manifestations.6,7,35

One such important distinction comes from the separa-

tion of the SIE into many- and one-electron components,4

of which, we will be only concerning ourselves with the

one-electron part due to the diﬃculty in writing down a

clear expression for the many-electron portion.

In fact, we already extensively analyzed the one-

electron SIE in a previous paper67 in which we used

mono- and dinuclear one-electron systems to probe the

various components on 74 diﬀerent DFAs. We also thor-

oughly characterized the basis set dependence of the

SIE and correlated it with the exponents of the applied

Gaussian-type orbitals, which impact the latters’ diﬀuse-

ness. We also demonstrated that the one-electron SIE is a

sizeable component in several of the most accurate DFAs

to date, and that there is an approximately linear rela-

tionship between the one-electron SIE and the nuclear

charge. However, our previous, relatively simple model

systems might be insuﬃcient to generalize our ﬁndings

for more realistic calculations. Therefore, we have set

about extending our information on one-electron SIE to

more complicated model systems, thus, hopefully extend-

ing these insights to real-world applications.

A critical task in computational chemistry is surveying

potential energy surfaces for global minima and transi-

tion states prior to modelling chemical or physical prop-

erties. Just as the SIE can impact desired quantities

directly, e.g. overly red-shifted excitation energies and

spectra—see Refs [61] and[64] for reviews on DFT for ex-

cited states — it can also aﬀect them indirectly through

artiﬁcially stable or unstable geometries, e.g. failure to

predict a transition state. As the structural arrange-

ment of atoms often underpins fundamental chemical

and physical properties of molecules and solids, some-

times even subtle alterations of predicted geometries can

cause speciously-calculated properties. Through shifting

the topology of the potential energy surface, the SIE can

cause qualitative failures in the prediction of structures.

This is why robustness in particular is such a highly

prized feature of a DFA in many benchmark studies.3,63

Therefore, we ﬁnd it valuable to consider the one-electron

SIE in the context of diﬀerent geometries to provide more

boundaries around when our current methods could fail.

Additionally, how we have chosen to construct our one-

electron SIE datasets in our previous study has left us

with electronic structures corresponding to only the 1s

atomic orbital or MOs formed from 1s orbitals.67 As va-

lence orbitals are usually the most important for describ-

ing chemical bonding and changes of the electronic struc-

ture during a reaction, it follows that the SIE picture will

change in more realistic systems containing nuclei beyond

helium. Therefore, we have also sought to account for the

impact the SIE has on higher-lying orbitals. In this way,

we hope to incrementally build from the one-electron pic-

ture, a more realistic description of the SIE.

Herein, we present results and discussion into the ge-

ometry aspect of the SIE, going through each system and

the breakdown of errors within. Then, we visualise the

SIE with density-diﬀerence plots and touch on SIE ef-

fects when increasing the nuclear charge in our various

geometries. Finally, we connect the magnitude of the

one-electron SIE to higher-lying 2s and 2p atomic or-

bitals. These new ﬁndings will provide further insights

and data that can be used in the development of one-

electron SIE corrections or SIE-free DFAs.

A. Theoretical background

We present only a brief series of deﬁnitions of the SIE

and the breakdown into its components: a more detailed

discussion can be found in the theoretical details of our

previous study.67

As we consider only one-electron systems within the

Born-Oppenheimer Approximation without relativistic

eﬀects, HF Theory correctly gives the exact energy and

density for such systems. To account for the self-

interaction energy from eq. (1), a set of exchange in-

tegrals generate an energy equivalent in magnitude but

opposite in sign — the result is that these two quantities

vanish and the exact-energy expression for one-electron

systems becomes:

EHF

1el [ρ] = h[ρ],(2)

where his the expectation value of the one-electron

Hamiltonian — stemming from electronic kinetic and

electron-nucleus interaction energies — and ρis the exact

one-electron density. As many DFAs are not one-electron

SIE free and only yield an approximate density and total

energy we can express the SIE of such a functional as:

SIE[˜ρ, ρ] = EDF A[˜ρ]−EHF

1el [ρ]

=EDF A[˜ρ]−h[ρ]

=J[˜ρ] + EDF A

X[˜ρ] + EDF A

C[˜ρ] + h[˜ρ]−h[ρ],

(3)

where ˜ρis the approximate electron density for the

DFA, Jis the classic Coulomb repulsion energy, and

EDF A

Xand EDF A

Care the DFAs’ exchange and correla-

tion energies, respectively. We can then break the one-

electron SIE down into various component parts which

we brieﬂy detail below. Firstly, improper cancellation be-

tween the approximate DFA exchange and the Coulomb

energy leads to the exchange SIE:

ESIE

X[˜ρ] = EDF A

X[˜ρ] + J[˜ρ].(4)

Note that we have dropped the subscript 1el, as it is clear

that the context of our SIE discussion is the one-electron

case. As electron correlation for one electron systems

should be zero by deﬁnition, we can simply write the

correlation energy as the correlation SIE:

ESIE

C[˜ρ] = EDF A

C[˜ρ].(5)

As EDF A

Xand EDF A

Care approximate, applying them

during an SCF procedure causes an incorrect density, and

in turn an error from the formally exact one-electron term

h, which we deﬁned as the one-electron error (OEE):67

OEE[˜ρ, ρ] = h[˜ρ]−h[ρ].(6)

The sum of eqs 4, 5, & 6 returns the total SIE in eq. 3.

Not speciﬁc to the one-electron SIE, Sim, Burke, and

co-workers68,69 have broken down errors from DFAs into

density- and functional-based components. This is an

attempt to separate the error stemming from an incor-

rectly calculated density, through the self-consistent ﬁeld

(SCF) procedure, from the error that comes from a sin-

gle non-SCF DFA calculation — presupposing an accu-

rately/exactly calculated density. We expanded upon

this concept in our last paper67 to speciﬁcally cover the

one-electron SIE, so the functional error becomes:

SIEf unc[ρ] = EDF A[ρ]−EHF [ρ](7)

and the density-error is:

SIEdens[˜ρ, ρ] = EDF A[˜ρ]−EDF A[ρ].(8)

We would like to note these formulations of SIEf unc

and SIEdens are only relevant in the context of the one-

electron SIE and that they would need to be adapted to

apply to many electron systems; see Ref [70] for a recent

appraisal of density-corrected DFT. The sum of eqs 7 and

8 return the total one-electron SIE as deﬁned in eq. 3.

The OEE from eq. 6 is, thus, related to and a part of the

SIEdens from eq. 8.67

B. Computational details

Calculations were performed with QCHEM V5.3.171,

V5.2 in the case of Sections II A 4 and II B, using unre-

stricted Kohn-Sham DFT and HF algorithms with the

decontracted aug-cc-pVQZ basis set.72,73 This approach

is the same as our previous study, so we refrain from

a basis-set dependence study: see Ref [67] for details.

In order to calculate some breakdowns of the SIE (see

eqs 7 and 8) some of the DFT and HF calculations

were followed by non-SCF procedures, where appropri-

ate. For numerical quadrature we used an unpruned grid

of 99 radial spheres and 590 angular spheres with stan-

dard convergence thresholds. There were several cases of

poor convergence, whereby we used standard convergence

techniques as well as stability analysis, level shifting, and

various damping variants.

To investigate the SIE’s eﬀect on a range of geome-

tries we chose several common nuclear arrangements pre-

sented in Fig. 1. We have split them into 3 cate-

gories: one-dimensional strings of 2, 3, 4, or 5 nuclei,

dubbed “Linear 1”, ‘Linear 2” etc.; two-dimensional

shapes including a Triangle,Square,Pentagon, and

Linear 3

Linear 4

Linear 5

Linear 2

Triangle

Square

Hexagon

Pentahedron

Octahedron

Tetrahedron

Pentagon

FIG. 1: Geometries investigated in this study. Polygons

and polyhedra are regular; linear systems possess D∞h

symmetry.

Hexagon; and three-dimensional Tetrahedron,Pen-

tahedron, and Octahedron geometries, again labelled

in bold in the following discussion. These shapes were

chosen as model systems because they can be regular,

highly symmetric arrangements with each nucleus being

equidistant to all its nearest neighbors. The equilateral

side length in all polygons and polyhedra is denoted here

as “r”. For the strings of nuclei all internuclear distances

between nearest neighbors were the same and also de-

noted as “r”. By introducing such symmetry we can char-

acterize the delocalized nature of the one-electron SIE

through an idealized set of parameters. In order to study

the SIE exactly, all systems possess exactly one electron.

All atomic positions are predominately hydrogen nuclei

— except for Section (II A 4) which considers the eﬀect

of higher nuclear charge. The rvalue was increased to

create dissociation curves for all the structures from 0.1

to 10 Å in 100 points with equal spacing. In particular,

we only calculated these points for the hydrogen nuclei,

for nuclei helium to carbon we calculated rvalues from

0.5 to 10 Å in 20 equally spaced points.

Due to the unusual nature of our test systems there

were severe convergence issues for several DFAs we oth-

erwise might have included. That being said, the purpose

of our study was not to rank a large number of DFAs, like

we have done in our previous work, but to exemplify SIE-

related behavior for geometries and higher-lying occupied

orbitals. For that reason, we limited our selection to sev-

eral common and/or unique functionals that had been in-

sightful in our previous study.67 DFAs tested on the var-

ious geometries were: SVWN5,74,75 BLYP,9–11 PBE,76

TPSS,77 B97M-V,78 B3LYP,79,80 PBE0,81,82 SCAN0,83

BHLYP,53 B75LYP (75 % Fock exchange), and ωB97X-

V.84 In the latter, the nonlocal dispersion correction of

the VV10 type85 was used in its full-SCF implementa-

tion; see Ref. [86] for a discussion on how VV10 can be

applied for this and related functionals. We also note that

we demonstrated in our previous study, how the VV10

correction itself contributed to the SIE;67 this aspect of

ωB97X-V will therefore not be discussed again. Note we

only used SVWN5, BLYP, PBE, and B97M-V for Section

II A 4.

Higher-orbital calculations for Section II B were

achieved with the maximum overlap method (MOM).87

In this case, a set of ground state orbitals based on the

DFA of choice formed the basis for an MOM calcula-

tion. This procedure was adopted to calculate the to-

tal energies of mononuclear one-electron systems con-

taining the nuclei from hydrogen to carbon when the

single electron occupies the 2s or 2p orbitals. An ini-

tial calculation with the DFA of choice was done to

generate the orbitals, then the occupied orbital was

swapped with the higher lying orbital of choice. Fol-

lowing this, an MOM calculation was started with over-

lap being maximized with the initial guess orbitals.

DFAs tested in this part were: SVWN5, BLYP, PBE,

revPBE,88 RPBE,89 TPSS, SCAN,90 B3LYP, PBE0,

M11,91 MN15,92 CAM-B3LYP,93 N12-SX,94 ωB97X-V,

B2PLYP,95 ωB97M(2).96 Note that the last two are dou-

ble hybrids95,97 and that for one-electron systems, only

their hybrid portions are relevant. Again, the choice of

DFAs was inspired by our previous ﬁndings.67 Most of

these calculations converged slightly better, which means

we could include more DFAs than in our geometry study;

exceptions were some gaps for RPBE, TPSS, SCAN,

M11, and MN15, for which the MOM procedure did not

always converge. Those DFAs were still included where

they worked to allow at least for a qualitative study of

them.

II. RESULTS AND DISCUSSION

A. Geometry and the self-interaction error

In the context of our previous work,67 the most logical

extension from our original dimer study is to investigate

the eﬀect a third nucleus has on the SIE. This is achieved

with a heterolytic dissociation curve of the asymmetric,

linear H2+

3trimer: the only system with such asymmetry

to be considered in our work. Following this, we conduct

and analysis of the SIE and its components, speciﬁcally of

the Linear 3 and Triangle geometries. Then, we sim-

plify the discussion by using mean absolute percentage

error (MAPE) values to more broadly cover the remain-

ing systems and DFAs studied. In this work, the MAPE

is deﬁned as:

MAP E =1

iSIEDF A

EHF

1el i

×100% ,(9)

which is an averaged sum of SIE values for each dis-

tance ifor a given system as a percentage of the electronic

HF energy, which is identical to EHF

1el , for that same sys-

tem and distance. While it would have been desirable

to analyze MAPEs for the entire range of runtil the HF

dissociation limit has been reached, severe convergence

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One-electronself-interactionerroranditsrelationshiptogeometryandhigherorbitaloccupationDaleR.LonsdaleandLarsGoerigkSchoolofChemistry,TheUniversityofMelbourne,Victoria3010,AustraliaDensityFunctionalTheory(DFT)seesprominentuseincomputationalchemistryandphysics,however,problemsduetotheself-interaction...

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One-electron self-interaction error and its relationship to geometry and higher orbital occupation Dale R. Lonsdale and Lars Goerigk

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